First slide

Pair of straight lines

Question

If one of the lines represented by $2 x^{2} + 2 hxy + 3 y^{2} = 0$ be perpendicular to one of the lines given by $3 x^{2} - 2 h^{'} xy + 2 y^{2} = 0$ , then

Moderate

A

$h - h^{'} = 0$
B

$h + h^{'} = 0$
C

$3 h = 2 h^{'}$
D

$2 h = 3 h^{'}$

Solution

Suppose that $y = mx$ is one of the lines represented by the equation $2 x^{2} + 2 hxy + 3 y^{2} = 0$
It implies that ${3m}^{2} + 2 hm + 3 = 0$ …… (1)
Since, equation of the line perpendicular to the line $y = mx$ is $y = - \frac{1}{m} x$
Suppose that $y = - \frac{1}{m} x$ is one of the lines represented by the equation $3 x^{2} - 2 h^{'} xy + 2 y^{2} = 0$
It implies that $2 m^{2} - 2 h^{'} m + 3 = 0$ …… (2)
Comparing the above two conditions, $\Rightarrow h + h^{'} = 0 (since, m \neq 0)$

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